=6n
∀
n,
and Y
2n-1
=n. Both subsequences converge
to
+
∞
; therefore Lim Sup Y
n
=Lim Inf Y
n
=
+
∞
and the subsequential limit is {
+
∞
}.

4d) The subsequence can be written as Z
2n
=n
n
which converges to
+
∞
∀
n,
and Z
2n-1
= -n
n
which converges to -
∞
. Lim Sup Z
n
=
+
∞
and Lim Inf Z
n
=
−
∞
and the subsequential
limit is {
−
∞,
+
∞
}.
6a) False; S
n
=n* (-1)
n
S
n
={ -1, 2, -3, 4, -5, 6, ….} for all n. The sequence oscillates but does not
converge.
6b) True; By definition 4.4.9, If a sequence oscillates then the Sup and Inf of the sequence are
not equal. This implies that the sequence can not converge because all subsequences would
converge to the same limit.
6c) False; Let S
n
= n for all n
∈
N
. The sequence diverges to
+
∞
but it is not an oscillating
sequence.
7a) True; This stems from definition 4.3.9, Lemma 4.3.10 & 4.3.11. Also Theorem 4.4.7.
Every convergent sequence is a Cauchy Sequence. Every Cauchy Sequence is bounded.
Every bounded sequence has a convergent subsequence.
7b) False; Let S
n
= n for all n
∈
N
. Then S
n
is monotone S
n
≤S
n+1
the sequence is increasing.
This monotone sequence is not bounded above
By theorem 4.4.8, every unbounded sequence
contains a monotone subsequence that has either
−
∞
∨+
∞
as a limit. If the sequence is not
bounded, then the subsequence is not bounded.
Section 5.1
4) For
ϵ
>
0
∧
δ
>
0,
|x
2
+2x-8| <1/4
|x+4||x-2|<1/4
Because | x+4| is an upper bound, let |x-2|<1/4, then |x+4|=|x-2+6|
≤
|x-2|+6< 7
|x
2
+2x-8|=|x+4||x-2|< 7|x-2| and
δ
=
1
28
such that |x-2|<
δ
6a) For
ϵ
>
0
∧
δ
>
0,
| (x
2
-5x+1)-(-5)|= | x
2
-5x+6|=|x-2||x-3| <
ϵ
, for every 0<|x-3|<
δ
.
Let S
1
=1. Then 0<|x-3|<
δ
1
=1
|x-2|=|(x-3) +1|
≤
|x-3|+1
≤δ
1
+1 =2
Therefore | (x
2
-5x+1)-(-5)||=|x-2||x-3|<2|x-3|<
ϵ
where
δ
=
ϵ
8
.
6b) For
ϵ
>
0
∧
δ
>
0,
| (x
2
+3x+8)-(8)|= | x
2
+3x|=|x||x+3| <
ϵ
, for every 0<|x+3|<
δ
Let S
1
=1. Then 0<|x+3|<
δ
1
=1
|x|=|(x+3)-3|
≤
|(x+3)+|-3|<
δ
1
+3=4

Therefore | (x
2
+3x+8)-(8)||=|x||x+3|<4|x+3|<
ϵ
where
δ
=
ϵ
4
.

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- Fall '08
- Staff
- Limits, Limit of a sequence, Limit superior and limit inferior, lim sup, subsequence, Lim Sup Vn